The maximum degree of the rth power of a sparse random graph

Let Gʳn,p denote the rth power of the random graph Gn,p, where $p=c/n$ for a positive constant c. We prove that w.h.p. the maximum degree Δ(Gʳn,p)~ log nlog₍ᵣ₊₁₎n. Here log₍ₖ₎n indicates the repeated application of the log-function k times. So, for example, log₍₃₎n=logloglog n.